Advertisement

Approximating Small Balanced Vertex Separators in Almost Linear Time

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any \(\frac{2}{3}\le \alpha <1\) and any \(0<\varepsilon <1-\alpha \): If G contains an \(\alpha \)-separator of size K, then our algorithm finds an \((\alpha +\varepsilon )\)-separator of size \(\mathcal O(\varepsilon ^{-1}K^2\log ^{1+o(1)} n)\) in time \(\mathcal O(\varepsilon ^{-1}K^3m\log ^{2+o(1)} n)\) w.h.p. In particular, if \(K\in \mathcal O({\text {polylog}}\, n)\), then we obtain an \((\alpha +\varepsilon )\)-separator of size \(\mathcal O(\varepsilon ^{-1}{\text {polylog}}\, n)\) in time \(\mathcal O(\varepsilon ^{-1}m\,{\text {polylog}}\, n)\) w.h.p. The presented algorithm does not require knowledge of K.

Due to space restrictions, no proofs are included in this version of the paper; the full version with a lot of additional material can be found at http://disco.ethz.ch/publications/wads2017-vertexsep.pdf.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Seymour, P.D., Thomas, R.: A separator theorem for graphs with an excluded minor and its applications. In: STOC (1990)Google Scholar
  2. 2.
    Amir, E., Krauthgamer, R., Rao, S.: Constant factor approximation of vertex-cuts in planar graphs. In: STOC (2003)Google Scholar
  3. 3.
    Arora, S., Kale, S.: A combinatorial, primal-dual approach to semidefinite programs. In: STOC (2007)Google Scholar
  4. 4.
    Bui, T.N., Jones, C.: Finding good approximate vertex and edge partitions is NP-hard. Information Processing Letters 42(3), 153–159 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Djidjev, H.N.: A linear algorithm for partitioning graphs of fixed genus. Serdica 11(4), 369–387 (1985)MathSciNetGoogle Scholar
  6. 6.
    Even, G., Naor, J., Rao, S., Schieber, B.: Divide-and-conquer approximation algorithms via spreading metrics. In: FOCS (1995)Google Scholar
  7. 7.
    Even, G., Naor, J., Rao, S., Schieber, B.: Fast approximate graph partitioning algorithms. In: SODA (1997)Google Scholar
  8. 8.
    Feige, U., Hajiaghayi, M.T., Lee, J.R.: Improved approximation algorithms for minimum-weight vertex separators. In: STOC (2005)Google Scholar
  9. 9.
    Feige, U., Mahdian, M.: Finding small balanced separators. In: STOC (2006)Google Scholar
  10. 10.
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. Journal of Algorithms 5(3), 391–407 (1984)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gonzalez, J.E., Low, Y., Gu, H., Bickson, D., Guestrin, C.: Powergraph: distributed graph-parallel computation on natural graphs. In: OSDI (2012)Google Scholar
  12. 12.
    Ford Jr., L.R., Fulkerson, D.R.: Maximal flow through a network. Canadian Journal of Mathematics 8, 399–404 (1956)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kawarabayashi, K.I., Reed, B.A.: A separator theorem in minor-closed classes. In: FOCS (2010)Google Scholar
  14. 14.
    Leighton, F.T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46(6), 787–832 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36(2), 177–189 (1979)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Malewicz, G., Austern, M.H., Bik, A.J.C., Dehnert, J.C., Horn, I., Leiser, N., Czajkowski, G.: Pregel: a system for large-scale graph processing. In: SIGMOD (2010)Google Scholar
  17. 17.
    Marx, D.: Parameterized graph separation problems. Theoretical Computer Science 351(3), 394–406 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Menger, K.: Zur allgemeinen Kurventheorie. Fundamenta Mathematicae 10(1), 96–115 (1927)MATHGoogle Scholar
  19. 19.
    Reed, B.A., Wood, D.R.: A linear-time algorithm to find a separator in a graph excluding a minor. ACM Transactions on Algorithms 5(4) (2009)Google Scholar
  20. 20.
    Wulff-Nilsen, C.: Separator theorems for minor-free and shallow minor-free graphs with applications. In: FOCS (2011)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.ETH ZürichZürichSwitzerland

Personalised recommendations